How do you coach your team to understand the importance of a positive, solution-oriented mindset within IT versus a "department of no" mentality?

I find that leaning onto their own leadership strengths will elicit a lot. We often discount how people have advanced and arrived to where they are now and thus an dialog on the past initiatives they had led, the positive outcomes they witnessed, how they picked up or how they supported each other when they stumbled are real nuggets to elicit. These could come from large scale projects they led or weekend volunteer efforts they led. 

I worked for a very smart person once who told me --- "our job is not to say 'no'.  Our job is to say 'yes, and here's how.'  It's up to the business to decide whether the cost and the time balance out to the value we are presenting."    The only exception is if it is an actual, physical impossibility...which today, is much harder to come by.

Another very smart person showed me how to "say no in a yes way".  It's similar, but slightly different...especially when dealing in service desk/technical support.  Rather than saying "no, I can't/won't/am not authorized to do xyz", tell the customer what you can/will/do to help them -- even if it's helping THEM fill out a form to send to another department.  

People who say "no" get that reputation, and then customers stop coming.  When customer stop coming, you just worked yourself out of a job.  Be the team of "yes" and be valued!

I encourage use of the phrase "Everything is possible in IT, but it might not be possible very soon with current resources and priorities ... so let's talk about resources and priorities"